"God, by definition, is that for which no greater can be conceived. God exists in the understanding. If God exists in the understanding, we could imagine Him to be greater by existing in reality. Therefore, God must exist."

This statement doesn't prove God's existence. Only solipsism would validate that. And I can conceive of quite a few things better than a creepy judgmental father figure who allows suffering all over the world even though he supposedly has the power to stop it.

[T]here is an evident absurdity in pretending to demonstrate a matter of fact, or to prove it by any arguments a priori. Nothing is demonstrable, unless the contrary implies a contradiction. Nothing, that is distinctly conceivable, implies a contradiction. Whatever we conceive as existent, we can also conceive as non-existent. There is no being, therefore, whose non-existence implies a contradiction. Consequently there is no being, whose existence is demonstrable.
-- David Hume, taken from the "Criticisms" section of your link